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{{Short description|Conjecture in number theory}}
In [[Algebraic geometry|algebraic geometry]] and [[Number theory|number theory]], the '''torsion conjecture''' or '''uniform boundedness conjecture for torsion points''' for abelian varieties states that the [[Order (group theory)|order]] of the [[Torsion group|torsion group]] of an abelian variety over a number field can be bounded in terms of the [[Dimension of an algebraic variety|dimension of the variety]] and the number field. A stronger version of the conjecture is that the torsion is bounded in terms of the dimension of the variety and the degree of the number field. The torsion conjecture has been completely resolved in the case of elliptic curves.

==Elliptic curves==
{{Infobox mathematical statement
| name = Ogg's conjecture
| image = 
| caption = 
| field = [[Number theory]]
| conjectured by = Beppo Levi
| conjecture date = 1908
| first proof by = [[Biography:Barry Mazur|Barry Mazur]]
| first proof date = 1977–1978
| open problem = 
| known cases =
| implied by =
| equivalent to = 
| generalizations =
| consequences =
}}
From 1906 to 1911, Beppo Levi published a series of papers investigating the possible finite orders of points on elliptic curves over the rationals.{{sfn|Schappacher|Schoof|1996|pp=64–65}} He showed that there are infinitely many elliptic curves over the rationals with the following torsion groups:
* ''C''<sub>''n''</sub> with 1 ≤ ''n'' ≤ 10, where ''C''<sub>''n''</sub> denotes the [[Cyclic group|cyclic group]] of order ''n'';
* ''C''<sub>12</sub>;
* ''C''<sub>2n</sub> × ''C''<sub>2</sub> with 1 ≤ ''n'' ≤ 4, where × denotes the [[Direct sum of groups|direct sum]].
At the 1908 International Mathematical Congress in Rome, Levi conjectured that this is a complete list of torsion groups for elliptic curves over the rationals.{{sfn|Schappacher|Schoof|1996|pp=64–65}} The torsion conjecture for elliptic curves over the rationals was independently reformulated by {{harvs|txt|first=Trygve|last=Nagell|year=1952}} and again by {{harvs|txt|first=Andrew|last=Ogg|year=1971}}, with the conjecture becoming commonly known as '''Ogg's conjecture'''.{{sfn|Schappacher|Schoof|1996|pp=64–65}}

{{harvs|txt|first=Andrew|last=Ogg|year=1971}} drew the connection between the torsion conjecture for elliptic curves over the rationals and the theory of [[Classical modular curve|classical modular curve]]s.{{sfn|Schappacher|Schoof|1996|pp=64–65}} In the early 1970s, the work of Gérard Ligozat, Daniel Kubert, [[Biography:Barry Mazur|Barry Mazur]], and John Tate showed that several small values of ''n'' do not occur as orders of torsion points on elliptic curves over the rationals.{{sfn|Schappacher|Schoof|1996|pp=64–65}} {{harvs|txt|first=Barry|last=Mazur|authorlink=Barry Mazur|year=1977|year2=1978}} proved the full torsion conjecture for elliptic curves over the rationals. His techniques were generalized by {{harvtxt|Kamienny|1992}} and {{harvtxt|Kamienny|Mazur|1995}}, who obtained uniform boundedness for [[Quadratic field|quadratic field]]s and number fields of degree at most 8 respectively. Finally, {{harvs|txt|first=Loïc|last=Merel|authorlink=Loïc Merel|year=1996}} proved the conjecture for elliptic curves over any number field.{{sfn|Schappacher|Schoof|1996|pp=64–65}}

An effective bound for the size of the torsion group in terms of the degree of the number field was given by {{harvtxt|Parent|1999}}. A complete list of possible torsion groups has also been given for elliptic curves over quadratic number fields. There are substantial partial results for quartic and quintic number fields {{harv|Sutherland|2012}}.

==See also==
* [[Bombieri–Lang conjecture]]
* Uniform boundedness conjecture for preperiodic points
* [[Uniform boundedness conjecture for rational points]]

==References==
{{reflist}}

==Bibliography==
*{{cite journal | last=Kamienny | first=Sheldon | title=Torsion points on elliptic curves and <math>q</math>-coefficients of modular forms | year=1992 | journal=Inventiones Mathematicae | volume=109 | issue=2 | pages=221–229 | doi=10.1007/BF01232025 | mr=1172689 | bibcode=1992InMat.109..221K | s2cid=118750444 }}
*{{cite journal | last1=Kamienny | first1=Sheldon | last2=Mazur | first2=Barry |  title=Rational torsion of prime order in elliptic curves over number fields | others=With an appendix by A. Granville | journal=[[Astérisque]] | year=1995 | pages=81–100 | mr=1330929 | volume=228}}
*{{cite journal|last=Mazur|first=Barry|title=Modular curves and the Eisenstein ideal|volume=47|issue=1|pages=33–186|year=1977|doi=10.1007/BF02684339|mr=0488287|journal=[[Publications Mathématiques de l'IHÉS]]|s2cid=122609075|url=http://www.numdam.org/item/PMIHES_1977__47__33_0/}}
*{{citation|last=Mazur|first=Barry|title=Rational isogenies of prime degree|volume=44|issue=2|pages=129–162|year=1978|doi=10.1007/BF01390348|mr=0482230|journal=[[Inventiones Mathematicae]]|others=with appendix by Dorian Goldfeld|bibcode=1978InMat..44..129M|s2cid=121987166}}
*{{cite journal | last1=Merel | first1=Loïc |  title=Bornes pour la torsion des courbes elliptiques sur les corps de nombres | trans-title=Bounds for the torsion of elliptic curves over number fields | language=fr | doi=10.1007/s002220050059 |mr=1369424 | year=1996 | journal=[[Inventiones Mathematicae]] | volume=124 | issue=1 | pages=437–449 | bibcode=1996InMat.124..437M | s2cid=3590991 }}
*{{cite book |last=Nagell |first=Trygve |chapter=Problems in the theory of exceptional points on plane cubics of genus one |title=Den 11te Skandinaviske Matematikerkongress, Trondheim 1949, Oslo |publisher={{ill|Johan Grundt Tanum forlag|no}} |pages=71–76 |year=1952|oclc=608098404}}
*{{cite journal |last=Ogg |first=Andrew |year=1971 |title=Rational points of finite order on elliptic curves |journal=[[Inventiones Mathematicae]] |volume=22 |issue=2 |pages=105–111|doi=10.1007/BF01404654|bibcode=1971InMat..12..105O |s2cid=121794531 }}
*{{cite journal|last=Ogg |first=Andrew |title=Rational points on certain elliptic modular curves|journal=Proc. Symp. Pure Math. |volume=24 |pages=221–231 |year=1973|doi=10.1090/pspum/024/0337974 |series=Proceedings of Symposia in Pure Mathematics |isbn=9780821814246 }}
*{{cite journal | last=Parent | first=Pierre | title=Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres | trans-title=Effective bounds for the torsion of elliptic curves over number fields | language=fr | journal=Journal für die Reine und Angewandte Mathematik | year=1999 | volume=1999 | issue=506 | pages=85–116 | mr=1665681 | doi=10.1515/crll.1999.009 | arxiv=alg-geom/9611022 }}
*{{Citation
  | last1 = Schappacher
  | first1 = Norbert
  | last2 = Schoof
  | first2 = René
  | title = Beppo Levi and the arithmetic of elliptic curves
  | journal = [[The Mathematical Intelligencer]]
  | volume = 18
  | issue = 1
  | pages = 57–69
  | year = 1996
  | doi = 10.1007/bf03024818
  | mr = 1381581
  | zbl = 0849.01036
  | s2cid = 125072148
 | url = http://irma.math.unistra.fr/~schappa/NSch/Publications_files/1996_RSchNSch.pdf
}}
*{{cite web | last=Sutherland | first=Andrew V. | title=Torsion subgroups of elliptic curves over number fields | year=2012 | url=https://math.mit.edu/~drew/MazursTheoremSubsequentResults.pdf|website=math.mit.edu }}

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